Worksheet: Distribution of Molecular Speeds

In this worksheet, we will practice calculating the proportion of particles, in an ideal gas, that have a given speed using the Maxwell–Boltzmann distribution function.

Q1:

An incandescent light bulb is filled with neon gas. The gas that is in close proximity to the element of the bulb is at a temperature of 2 3 0 0 K. Determine the root-mean-square speed of neon atoms in close proximity to the element. Use a value of 20.2 g/mol for the molar mass of neon.

  • A 3 . 2 × 1 0 3 m/s
  • B 2 2 × 1 0 3 m/s
  • C 4 . 5 × 1 0 3 m/s
  • D 1 . 7 × 1 0 3 m/s
  • E 3 . 6 × 1 0 3 m/s

Q2:

Helium atoms in a gas that is at a temperature 𝑇 1 have a root-mean-square speed of 196 m/s. When the gas is heated until it becomes a plasma with a temperature 𝑇 2 , the root-mean-square speed of the helium atoms is 618 km/s. Use a value of 4.003 g/mol for the molar mass of helium.

Find 𝑇 1 .

Find 𝑇 2 .

  • A 6 . 1 3 × 1 0 7 K
  • B 6 3 . 7 × 1 0 7 K
  • C 6 . 5 9 × 1 0 7 K
  • D 6 . 1 7 × 1 0 7 K
  • E 3 . 6 8 × 1 0 7 K

Q3:

Using the approximation 𝑓 ( 𝑣 ) 𝑣 𝑓 ( 𝑣 ) Δ 𝑣 d for small Δ 𝑣 , estimate the fraction of nitrogen molecules at a temperature of 3 . 0 0 × 1 0 K that have speeds between 290 m/s and 291 m/s. A nitrogen molecule has a mass of 4 . 6 5 × 1 0 kg.

Q4:

Find the ratio 𝑓 𝑣 𝑓 ( 𝑣 ) p r m s for hydrogen gas at a temperature of 77.0 K. Use a molar mass of 2.02 g/mol for hydrogen gas.

Q5:

In a sample of a monatomic gas, a number of molecules 𝑛 have speeds that are within a very small range around the root-mean-square speed of atoms in the gas, 𝑣 . A number of molecules 𝑛 have speeds that are within the same very small range around a speed of 3 . 0 0 𝑣 . Determine the ratio of 𝑛 to 𝑛 .

Q6:

A sample of nitrogen is at a temperature of 3 0 1 5 K. 𝑁 has a molar mass of 28.00 g/mol.

What is the most probable speed of the nitrogen molecules?

What is the average speed of the nitrogen molecules?

What is the root-mean-square speed of the nitrogen molecules?

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